As the focal length gets longer, you will get closer to the subject. It will appear bigger in your viewfinder. With the subject bigger in your viewfinder, the normal shaking that occurs with hand holding a camera will be magnified. A faster shutter speed will lessen blur caused by camera shake. The general rule is that you should have a shutter speed faster than 1/focal length. If you are shooting at 300 mm, then the shutter speed should be 1/300 or faster.
If you are shooting on Kilamanjaro Safari truck where it is extremely bumpy, you should go even faster, probably 1/1000 at 300 mm.
If you cannot get to those shutter speeds and get a good exposure, then increase your ISO to compensate.
The 70-300 lenses are probably your best choice to stay within your budget.
The lens I have found best for all around shooting at Animal Kingdom is a 80/100-400 mm but that would cost you in the neighborhood of 1400 dollars new for a Canon or Nikon model. Used lenses can be found that are cheaper. They both have the advantage of having a built in compensator for camera shake.
I can use this on the Safari, the wildlife preserve, the maharajah trail, Festival of the Lion King. It doesn't work well for wide angle panorama shots which the 70-300 won't work well either.